describing data

Describing Data

• The canonical descriptive strategy is to describe the data in terms of their underlying distribution

• As usual, we have a p-dimensional data matrix with variables X1, …, Xp

• The joint distribution is P(X1, …, Xp)• The joint gives us complete information about the

variables• Given the joint distribution, we can answer any question

about the relationships among any subset of variables– are X2 and X5 independent?– generating approximate answers to queries for large

databases or selectivity estimation• Given a query (conditions that observations must satisfy),

estimate the fraction of rows that satisfy this condition (the selectivity of the query)

• These estimates are needed during query optimization • If we have a good approximation for the joint distribution of data,

we can use it to efficiently compute approximate selectivities

Graphical Models

• In the next 3-4 lectures, we will be studying graphical models

• e.g. Bayesian networks, Bayes nets, Belief nets, Markov networks, etc.

• We will study:– representation– reasoning– learning

• Materials based on upcoming book by Nir Friedman and Daphne Koller. Slides courtesy of Nir Friedman.

Probability Distributions

• Let X1,…,Xp be random variables

• Let P be a joint distribution over X1,…,Xp

If the variables are binary, then we need O(2p) parameters to describe P

Can we do better?• Key idea: use properties of independence

Independent Random Variables

• Two variables X and Y are independent if– P(X = x|Y = y) = P(X = x) for all values x,y– That is, learning the values of Y does not change

prediction of X

• If X and Y are independent then – P(X,Y) = P(X|Y)P(Y) = P(X)P(Y)

• In general, if X1,…,Xp are independent, then– P(X1,…,Xp)= P(X1)...P(Xp)

– Requires O(n) parameters

Conditional Independence

• Unfortunately, most of random variables of interest are not independent of each other

• A more suitable notion is that of conditional independence

• Two variables X and Y are conditionally independent given Z if– P(X = x|Y = y,Z=z) = P(X = x|Z=z) for all values x,y,z– That is, learning the values of Y does not change prediction

of X once we know the value of Z

– notation: I ( X , Y | Z )

Example: Naïve Bayesian Model

• A common model in early diagnosis:– Symptoms are conditionally independent given the

disease (or fault)

• Thus, if – X1,…,Xp denote whether the symptoms exhibited by

the patient (headache, high-fever, etc.) and – H denotes the hypothesis about the patients health

• then, P(X1,…,Xp,H) = P(H)P(X1|H)…P(Xp|H),

• This naïve Bayesian model allows compact representation

– It does embody strong independence assumptions

Modeling assumptions:Ancestors can affect descendants' genotype only by passing genetic materials through intermediate generations

Example: Family trees

Noisy stochastic process:

Example: Pedigree• A node represents

an individual’sgenotype

Homer

Bart

Marge

Lisa Maggie

Markov Assumption

• We now make this independence assumption more precise for directed acyclic graphs (DAGs)

• Each random variable X, is independent of its non-descendents, given its parents Pa(X)

• Formally,I (X, NonDesc(X) | Pa(X))

Descendent

Ancestor

Parent

Non-descendent

X

Y1 Y2

Non-descendent

Markov Assumption Example

• In this example:– I ( E, B )– I ( B, {E, R} )– I ( R, {A, B, C} | E )– I ( A, R | B,E )– I ( C, {B, E, R} | A)

Earthquake

Radio

Burglary

Alarm

Call

I-Maps

• A DAG G is an I-Map of a distribution P if all the Markov assumptions implied by G are satisfied by P(Assuming G and P both use the same set of random

variables)

Examples:

X Y

x y P(x,y)0 0 0.250 1 0.251 0 0.251 1 0.25

X Y

x y P(x,y)0 0 0.20 1 0.31 0 0.41 1 0.1

Factorization

• Given that G is an I-Map of P, can we simplify the representation of P?

• Example:

• Since I(X,Y), we have that P(X|Y) = P(X)• Applying the chain rule

P(X,Y) = P(X|Y) P(Y) = P(X) P(Y)

• Thus, we have a simpler representation of P(X,Y)

X Y

Factorization Theorem

From assumption:

)X(NonDesc)X(Pa}X,X{

}X,X{)X(Pa

ii1i,1

1i,1i

⊆−

⊆

−

−

K

K

∏=i

iip1 ))X(Pa|X(P)X,...,X(P

Thm: if G is an I-Map of P, then

∏ −=i

1i1ip1 )X,...,X|X(P)X,...,X(PProof:• By chain rule:

• wlog. X1,…,Xp is an ordering consistent with G

• Since G is an I-Map, I (Xi, NonDesc(Xi)| Pa(Xi))

• We conclude, P(Xi | X1,…,Xi-1) = P(Xi | Pa(Xi) )

))X(Pa|)X(Pa}X,X{,X(I ii1i,1i −−K• Hence,

Consequences

• We can write P in terms of “local” conditional probabilities

If G is sparse,– that is, |Pa(Xi)| < k ,

each conditional probability can be specified compactly

– e.g. for binary variables, these require O(2k) params.

representation of P is compact– linear in number of variables

Pause…Summary

We defined the following concepts• The Markov Independences of a DAG G

– I (Xi , NonDesc(Xi) | Pai )

• G is an I-Map of a distribution P– If P satisfies the Markov independencies implied by G

We proved the factorization theorem• if G is an I-Map of P, then

∏=i

iin1 )Pa|X(P)X,...,X(P

• Let Markov(G) be the set of Markov Independencies implied by G

• The factorization theorem shows

G is an I-Map of P

• We can also show the opposite:

Thm: G is an I-Map

of P

Conditional Independencies

∏=i

iin PaXPXXP )|(),...,( 1

∏=i


Proof (Outline)

Example:X

Y

Z

)|()()|()|()(

),(),,(

),|(XYPXP

XZPXYPXPYXPZYXP

YXZP ==

)|( XZP=

Implied Independencies

• Does a graph G imply additional independencies as a consequence of Markov(G)?

• We can define a logic of independence statements

• Some axioms:– I( X ; Y | Z ) I( Y; X | Z )

– I( X ; Y1, Y2 | Z ) I( X; Y1 | Z )

d-seperation

• A procedure d-sep(X; Y | Z, G) that given a DAG G, and sets X, Y, and Z returns either yes or no

• Goal: d-sep(X; Y | Z, G) = yes iff I(X;Y|Z) follows from

Markov(G)

Paths

• Intuition: dependency must “flow” along paths in the graph

• A path is a sequence of neighboring variables

Examples:• R E A B• C A E R

Earthquake

Radio

Burglary

Alarm

Call

Paths

• We want to know when a path is– active -- creates dependency between end nodes– blocked -- cannot create dependency end nodes

• We want to classify situations in which paths are active.

Blocked Unblocked

E

R A

E

R A

Path Blockage

Three cases:– Common cause

–

–

Blocked Active

Blocked Unblocked

E

C

A

E

C

A

Path Blockage


– Intermediate cause

–

Blocked Active

Blocked Unblocked

E B

A

C

E B

A

CE B

A

C

Path Blockage


– Intermediate cause

– Common Effect

Blocked Active

Path Blockage -- General Case

A path is active, given evidence Z, if• Whenever we have the configuration

B or one of its descendents are in Z

• No other nodes in the path are in Z

A path is blocked, given evidence Z, if it is not active.

A C

B

A

– d-sep(R,B)?

Example

E B

C

R

– d-sep(R,B) = yes– d-sep(R,B|A)?

Example

E B

A

C

R

– d-sep(R,B) = yes– d-sep(R,B|A) = no– d-sep(R,B|E,A)?

Example

E B

A

C

R

d-Separation

• X is d-separated from Y, given Z, if all paths from a node in X to a node in Y are blocked, given Z.

• Checking d-separation can be done efficiently (linear time in number of edges)– Bottom-up phase:

Mark all nodes whose descendents are in Z– X to Y phase:

Traverse (BFS) all edges on paths from X to Y and check if they are blocked

Soundness

Thm: • If

– G is an I-Map of P– d-sep( X; Y | Z, G ) = yes

• then– P satisfies I( X; Y | Z )

Informally,• Any independence reported by d-separation is

satisfied by underlying distribution

Completeness

Thm: • If d-sep( X; Y | Z, G ) = no• then there is a distribution P such that

– G is an I-Map of P– P does not satisfy I( X; Y | Z )

Informally,• Any independence not reported by d-separation

might be violated by the underlying distribution• We cannot determine this by examining the

graph structure alone

I-Maps revisited

• The fact that G is I-Map of P might not be that useful

• For example, complete DAGs– A DAG is G is complete if we cannot add an arc without

creating a cycle

• These DAGs do not imply any independencies• Thus, they are I-Maps of any distribution

X1

X3

X2

X4

X1

X3

X2

X4

Minimal I-Maps

A DAG G is a minimal I-Map of P if• G is an I-Map of P• If G’ G, then G’ is not an I-Map of P

Removing any arc from G introduces (conditional) independencies that do not hold in P

Minimal I-Map Example

• If is a minimal I-Map

• Then, these are not I-Maps:

X1

X3

X2

X4

X1

X3

X2

X4

X1

X3

X2

X4

X1

X3

X2

X4

X1

X3

X2

X4

Constructing minimal I-Maps

The factorization theorem suggests an algorithm

• Fix an ordering X1,…,Xn

• For each i, – select Pai to be a minimal subset of {X1,…,Xi-1 },

such that I(Xi ; {X1,…,Xi-1 } - Pai | Pai )

• Clearly, the resulting graph is a minimal I-Map.

Non-uniqueness of minimal I-Map

• Unfortunately, there may be several minimal I-Maps for the same distribution– Applying I-Map construction procedure with different orders

can lead to different structures

E B

A

C

R

Original I-Map

E B

A

C

R

Order: C, R, A, E, B

Choosing Ordering & Causality

• The choice of order can have drastic impact on the complexity of minimal I-Map

• Heuristic argument: construct I-Map using causal ordering among variables

• Justification?– It is often reasonable to assume that graphs of causal

influence should satisfy the Markov properties.

P-Maps

• A DAG G is P-Map (perfect map) of a distribution P if– I(X; Y | Z) if and only if

d-sep(X; Y |Z, G) = yes

Notes:• A P-Map captures all the independencies in the

distribution• P-Maps are unique, up to DAG equivalence

P-Maps

• Unfortunately, some distributions do not have a P-Map

• Example:

• A minimal I-Map:

• This is not a P-Map since I(A;C) but d-sep(A;C) = no

⎪⎩

⎪⎨⎧

=⊕⊕

=⊕⊕=

1if61

0if121

),,(CBA

CBACBAP

A B

C

Bayesian Networks

• A Bayesian network specifies a probability distribution via two components:

– A DAG G– A collection of conditional probability distributions

P(Xi|Pai)

• The joint distribution P is defined by the factorization

• Additional requirement: G is a minimal I-Map of P

∏=i


Summary

• We explored DAGs as a representation of conditional independencies:

– Markov independencies of a DAG– Tight correspondence between Markov(G) and the

factorization defined by G– d-separation, a sound & complete procedure for

computing the consequences of the independencies– Notion of minimal I-Map– P-Maps

• This theory is the basis for defining Bayesian networks

Markov Networks

• We now briefly consider an alternative representation of conditional independencies

• Let U be an undirected graph

• Let Ni be the set of neighbors of Xi

• Define Markov(U) to be the set of independenciesI( Xi ; {X1,…,Xn} - Ni - {Xi } | Ni )

• U is an I-Map of P if P satisfies Markov(U)

Example

This graph implies that• I(A; C | B, D )• I(B; D | A, C )

• Note: this example does not have a directed P-Map

A

D

B

C

Markov Network Factorization

Thm: if

• P is strictly positive, that is P(x1, …, xn ) > 0 for all assignments

then• U is an I-Map of Pif and only if• there is a factorization

where C1, …, Ck are the maximal cliques in U

Alternative form:

∏=i

in CfXXP )(),,( 1 K

∑= i

iCg

n1 eZ1

XXP)(

),,( K

Relationship between Directed & Undirected Models

Chain Graphs

Directed Graphs

UndirectedGraphs

CPDs

• So far, we focused on how to represent independencies using DAGs

• The “other” component of a Bayesian networks is the specification of the conditional probability distributions (CPDs)

• We start with the simplest representation of CPDs and then discuss additional structure

Tabular CPDs

• When the variable of interest are all discrete, the common representation is as a table:

• For example P(C|A,B) can be represented by

A B P(C = 0 | A, B) P(C = 1 | A, B)

0 0 0.25 0.750 1 0.50 0.501 0 0.12 0.881 1 0.33 0.67

Tabular CPDs

Pros:• Very flexible, can capture any CPD of discrete

variables• Can be easily stored and manipulated

Cons:• Representation size grows exponentially with

the number of parents!• Unwieldy to assess probabilities for more than

few parents

Structured CPD

• To avoid the exponential blowup in representation, we need to focus on specialized types of CPDs

• This comes at a cost in terms of expressive power

• We now consider several types of structured CPDs

Causal Independence

• Consider the following situation

• In tabular CPD, we need to assess the probability of fever in eight cases

• These involve all possible interactions between diseases

• For three disease, this might be feasible….For ten diseases, not likely….

Disease 1 Disease 3Disease 2

Fever

Causal Independence

• Simplifying assumption:– Each disease attempts to cause fever, independently of the

other diseases– The patient has fever if one of the diseases “succeeds”

• We can model this using a Bayesian network fragment

Fever

Fever 1 Fever 3Fever 2

Disease 1 Disease 3Disease 2

OR gate

F = or(SF,F1,F2,F3)Hypothetical variables

“Fever caused by Disease i”

SpontenuousFever

Noisy-Or CPD

• Models P(X|Y1,…,Yk), X, Y1,…, Yk are all binary

• Paremeters:– pi -- probability of X = 1 due to Yi = 1

– p0 -- probability of X = 1 due to other causes

• Plugging these in the model we get

),...,|0(1),...,|1(

)1()1(),...,|0(

11

01

kk

i

Yik

YYXPYYXP

ppYYXP i

=−==

−−== ∏

Noisy-Or CPD

• Benefits of noisy-or– “Reasonable” assumptions in many domains

• e.g., medical domain

– Few parameters. – Each parameter can be estimated independently of

the others

• The same idea can be extended to other functions:noisy-max, noisy-and, etc.

• Frequently used in large medical expert systems

Context Specific Independence

• Consider the following examples:

• Alarm sound depends on– Whether the alarm

was set before leaving the house– Burglary– Earthquake

• Arriving on time depends on – Travel route– The congestion on the two

possible routes

Set EarthquakeBurglary

Alarm

TravelRoute

Route 2traffic

Route 1traffic

Arrive ontime

Context-Specific Independence

• In both of these example we have context-specific indepdencies (CSI)– Independencies that depends on a particular value of

one or more variables

• In our examples:– Ind( A ; B, E | S = 0 )

Alarm sound is independent of B and E when the alarm is not set

– Ind( A ; R2 | T = 1 )Arrival time is independent of traffic on route 2 if we choose to travel on route 1

Representing CSI

• When we have such CSI, P(X | Y1,…,Yk ) is the same for several values of Y1,…,Yk

• There are many ways of representing these regularities

• A natural representation: decision trees– Internal nodes: tests on parents– Leaves: probability distributions on X

• Evaluate P(X | Y1,…,Yk ) by traversing tree

S

B

.0

E .8

.1 .7

0 1

0 1

0 1

Detecting CSI

• Given evidence on some nodes, we can identify the “relevant” parts of the trees– This consists of the paths in the tree that are consistent with

context

• Example – Context S = 0– Only one path of tree is relevant

• A parent is independent given the context if it does not appear on oneof the relevant paths

S

B

.0

E .8

.1 .7

0 1

0 1

0 1

Decision Tree CPDs

Benefits• Decision trees offer a flexible and intuitive

language to represent CSI • Incorporated into several commercial tools for

constructing Bayesian networks

Comparison to noisy-or• Noisy-or CPDs require full trees to represent• General decision tree CPDs cannot be

represented by noisy-or

Inference in Bayesian Inference in Bayesian NetworksNetworks

Inference

• We now have compact representations of probability distributions:– Bayesian Networks– Markov Networks

• Network describes a unique probability distribution P

• How do we answer queries about P?

• We use inference as a name for the process of computing answers to such queries

Queries: Likelihood

• There are many types of queries we might ask. • Most of these involve evidence

– An evidence e is an assignment of values to a set E variables in the domain

– Without loss of generality E = { Xk+1, …, Xn }

• Simplest query: compute probability of evidence

• This is often referred to as computing the likelihood of the evidence

∑ ∑=1x

1 ),,,( )(kx

kxxPP ee KK

Queries: A posteriori belief

• Often we are interested in the conditional probability of a variable given the evidence

• This is the a posteriori belief in X, given evidence e

• A related task is computing the term P(X, e) – i.e., the likelihood of e and X = x for values of X – we can recover the a posteriori belief by

∑′

′==

==

x

xXPxXP

xXP),(

),()|(

ee

e

)(),(

)|(eP

eXPeXP =

A posteriori belief

This query is useful in many cases:• Prediction: what is the probability of an outcome

given the starting condition– Target is a descendent of the evidence

• Diagnosis: what is the probability of disease/fault given symptoms– Target is an ancestor of the evidence

• As we shall see, the direction between variables does not restrict the directions of the queries– Probabilistic inference can combine evidence form all

parts of the network

Queries: A posteriori joint

• In this query, we are interested in the conditional probability of several variables, given the evidence

P(X, Y, … | e )

• Note that the size of the answer to query is exponential in the number of variables in the joint

Queries: MAP

• In this query we want to find the maximum a posteriori assignment for some variable of interest (say X1,…,Xl )

• That is, x1,…,xl maximize the probability

P(x1,…,xl | e)

• Note that this is equivalent to maximizing

P(x1,…,xl, e)

Queries: MAP

We can use MAP for:• Classification

– find most likely label, given the evidence

• Explanation – What is the most likely scenario, given the evidence

Queries: MAP

Cautionary note:• The MAP depends on the set of variables• Example:

– MAP of X– MAP of (X, Y)

x y P(x,y)

0 0 0.35

0 1 0.05

1 0 0.3

1 1 0.3

Complexity of Inference

Thm:Computing P(X = x) in a Bayesian network is NP-hard

Not surprising, since we can simulate Boolean gates.

Hardness

• Hardness does not mean we cannot solve inference– It implies that we cannot find a general procedure

that works efficiently for all networks– For particular families of networks, we can have

provably efficient procedures

Approaches to inference

•Exact inference – Inference in Simple Chains– Variable elimination– Clustering / join tree algorithms

•Approximate inference– Stochastic simulation / sampling methods– Markov chain Monte Carlo methods– Mean field theory

Inference in Simple Chains

How do we compute P(X2)?

X1 X2

∑∑ ==11

)|()(),()( 121212xx

xxPxPxxPxP

Inference in Simple Chains (cont.)

How do we compute P(X3)?

• we already know how to compute P(X2)...

X1 X2

∑∑ ==22

)|()(),()( 232323xx

xxPxPxxPxP

X3

∑∑ ==11

)|()(),()( 121212xx

xxPxPxxPxP


How do we compute P(Xn)?

• Compute P(X1), P(X2), P(X3), …

• We compute each term by using the previous one

∑ ++ =ix

iiii xxPxPxP )|()()( 11

X1 X2 X3Xn

...

Complexity:

• Each step costs O(|Val(Xi)|*|Val(Xi+1)|) operations

• Compare to naïve evaluation, that requires summing over joint values of n-1 variables


• Suppose that we observe the value of X2 =x2

• How do we compute P(X1|x2)?– Recall that we it suffices to compute P(X1,x2)

X1 X2

)()|(),( 11221 xPxxPxxP =


• Suppose that we observe the value of X3 =x3

• How do we compute P(X1,x3)?

• How do we compute P(x3|x1)?

X1 X2

)|()(),( 13131 xxPxPxxP =

X3

∑

∑∑=

==

2

22

)|()|(

),|()|()|,()|(

2312

2131213213

x

xx

xxPxxP

xxxPxxPxxxPxxP


• Suppose that we observe the value of Xn =xn

• How do we compute P(X1,xn)?

• We compute P(xn|xn-1), P(xn|xn-2), … iteratively

X1 X2

)|()(),( 111 xxPxPxxP nn =

X3

∑

∑

++

+

=

=+

i

i

xinii

xiniin

xxPxxP

xxxPxxP

)|()|(

)|,()|(

11

11

Xn...


• Suppose that we observe the value of Xn =xn

• We want to find P(Xk|xn )

• How do we compute P(Xk,xn )?

• We compute P(Xk ) by forward iterations

• We compute P(xn | Xk ) by backward iterations

X1 X2

)|()(),( knknk xxPxPxxP =

Xk Xn......

Elimination in Chains

• We now try to understand the simple chain example using first-order principles

• Using definition of probability, we have

∑∑∑∑=d c b a

edcbaPeP ),,,,()(

A B C ED


• By chain decomposition, we get

A B C ED

∑∑∑∑

∑∑∑∑=

=

d c b a

d c b a

dePcdPbcPabPaP

edcbaPeP

)|()|()|()|()(

),,,,()(


• Rearranging terms ...

A B C ED

∑∑∑ ∑

∑∑∑∑=

=

d c b a

d c b a

abPaPdePcdPbcP

dePcdPbcPabPaPeP

)|()()|()|()|(

)|()|()|()|()()(


• Now we can perform innermost summation

• This summation, is exactly the first step in the forward iteration we describe before

A B C ED

∑∑∑

∑∑∑ ∑=

=

d c b

d c b a

bpdePcdPbcP

abPaPdePcdPbcPeP

)()|()|()|(

)|()()|()|()|()(

X


• Rearranging and then summing again, we get

A B C ED

∑∑

∑∑ ∑

∑∑∑

=

=

=

d c

d c b

d c b

cpdePcdP

bpbcPdePcdP

bpdePcdPbcPeP

)()|()|(

)()|()|()|(

)()|()|()|()(

X X

Elimination in Chains with Evidence

• Similarly, we understand the backward pass

• We write the query in explicit form

A B C ED

∑∑∑

∑∑∑=

=

b c d

b c d

dePcdPbcPabPaP

edcbaPeaP

)|()|()|()|()(

),,,,(),(


• Eliminating d, we get

A B C ED

∑∑

∑∑ ∑

∑∑∑

=

=

=

b c

b c d

b c d

cePbcPabPaP

dePcdPbcPabPaP

dePcdPbcPabPaPeaP

)|()|()|()(

)|()|()|()|()(

)|()|()|()|()(),(

X


• Eliminating c, we get

A B C ED

∑

∑ ∑

∑∑

=

=

=

b

b c

b c

bepabPaP

cePbcPabPaP

cePbcPabPaPeaP

)|()|()(

)|()|()|()(

)|()|()|()(),(

XX


• Finally, we eliminate b

A B C ED

)|()(

)|()|()(

)|()|()(),(

aePaP

bepabPaP

bepabPaPeaP

b

b

=

=

=

∑

∑

XXX

Variable Elimination

General idea:• Write query in the form

• Iteratively– Move all irrelevant terms outside of innermost sum– Perform innermost sum, getting a new term– Insert the new term into the product

∑ ∑∑∏=kx x x i

iin paxPXP3 2

)|(),( Le

A More Complex Example

Visit to Asia

Smoking

Lung CancerTuberculosis

Abnormalityin Chest

Bronchitis

X-Ray Dyspnea

• “Asia” network:

V S

LT

A B

X D

),|()|(),|()|()|()|()()( badPaxPltaPsbPslPvtPsPvP

• We want to compute P(d)• Need to eliminate: v,s,x,t,l,a,b

Initial factors

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: v,s,x,t,l,a,b

Initial factors

Eliminate: v

Note: fv(t) = P(t)In general, result of elimination is not necessarily a probability term

Compute: ∑=v

v vtPvPtf )|()()(

),|()|(),|()|()|()()( badPaxPltaPsbPslPsPtfv⇒

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: s,x,t,l,a,b

• Initial factors

Eliminate: s

Summing on s results in a factor with two arguments fs(b,l)In general, result of elimination may be a function of several variables

Compute: ∑=s

s slPsbPsPlbf )|()|()(),(


),|()|(),|(),()( badPaxPltaPlbftf sv⇒

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: x,t,l,a,b

• Initial factors

Eliminate: x

Note: fx(a) = 1 for all values of a !!

Compute: ∑=x

x axPaf )|()(



),|(),|()(),()( badPltaPaflbftf xsv⇒

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: t,l,a,b

• Initial factors

Eliminate: t

Compute: ∑=t

vt ltaPtflaf ),|()(),(




),|(),()(),( badPlafaflbf txs⇒

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: l,a,b

• Initial factors

Eliminate: l

Compute: ∑=l

tsl laflbfbaf ),(),(),(




),|(),()(),( badPlafaflbf txs⇒

),|()(),( badPafbaf xl⇒

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: b

• Initial factors

Eliminate: a,bCompute:

∑∑ ==b

aba

xla dbfdfbadpafbafdbf ),()(),|()(),(),(




),|()(),( badPafbaf xl⇒),|(),()(),( badPlafaflbf txs⇒

)(),( dfdbf ba ⇒⇒

Variable Elimination

• We now understand variable elimination as a sequence of rewriting operations

• Actual computation is done in elimination step

• Exactly the same computation procedure applies to Markov networks

• Computation depends on order of elimination

∑=x

kxkx yyxfyyf ),,,('),,( 11 KK

∏=

=m

ilikx i

yyxfyyxf1

,1,1,11 ),,(),,,(' KK

Complexity of variable elimination

• Suppose in one elimination step we compute

This requires • multiplications

– For each value for x, y1, …, yk, we do m multiplications

• additions

– For each value of y1, …, yk , we do |Val(X)| additions

Complexity is exponential in number of variables in the intermediate factor!

∏⋅⋅i

iYXm )Val()Val(

∏⋅i

iYX )Val()Val(

Understanding Variable Elimination

• We want to select “good” elimination orderings that reduce complexity

• We start by attempting to understand variable elimination via the graph we are working with

• This will reduce the problem of finding good ordering to graph-theoretic operation that is well-understood

Undirected graph representation

• At each stage of the procedure, we have an algebraic term that we need to evaluate

• In general this term is of the form:

where Zi are sets of variables

• We now plot a graph where there is undirected edge X--Y if X,Y are arguments of some factor

– that is, if X,Y are in some Zi

• Note: this is the Markov network that describes the probability on the variables we did not eliminate yet

∑ ∑∏=1

)(),,( 1y y i

ikn

fxxP iZKK

Chordal Graphs

• elimination ordering undirected chordal graph

Graph:• Maximal cliques are factors in elimination• Factors in elimination are cliques in the graph• Complexity is exponential in size of the largest

clique in graph

LT

A B

X

V S

D

V S

LT

A B

X D

Induced Width

• The size of the largest clique in the induced graph is thus an indicator for the complexity of variable elimination

• This quantity is called the induced width of a graph according to the specified ordering

• Finding a good ordering for a graph is equivalent to finding the minimal induced width of the graph

General Networks

• From graph theory:Thm:• Finding an ordering that minimizes the

induced width is NP-HardHowever,• There are reasonable heuristic for finding

“relatively” good ordering• There are provable approximations to the best

induced width• If the graph has a small induced width, there

are algorithms that find it in polynomial time

Elimination on Trees

• Formally, for any tree, there is an elimination ordering with induced width = 1

Thm• Inference on trees is linear in number of

variables

PolyTrees

• A polytree is a network where there is at most one path from one variable to another

Thm:• Inference in a polytree is linear in the

representation size of the network– This assumes tabular CPT representation

A

CB

D E

F G

H

Approaches to inference

•Exact inference – Inference in Simple Chains– Variable elimination– Clustering / join tree algorithms

•Approximate inference– Stochastic simulation / sampling methods– Markov chain Monte Carlo methods– Mean field theory

Stochastic simulation

•Suppose you are given values for some subset of the variables, G, and want to infer values for unknown variables, U

•Randomly generate a very large number of instantiations from the BN– Generate instantiations for all variables – start at root

variables and work your way “forward”

•Only keep those instantiations that are consistent with the values for G

•Use the frequency of values for U to get estimated probabilities

•Accuracy of the results depends on the size of the sample (asymptotically approaches exact results)

Markov chain Monte Carlo methods

• So called because– Markov chain – each instance generated in the sample

is dependent on the previous instance– Monte Carlo – statistical sampling method

• Perform a random walk through variable assignment space, collecting statistics as you go– Start with a random instantiation, consistent with

evidence variables– At each step, for some nonevidence variable, randomly

sample its value, consistent with the other current assignments

• Given enough samples, MCMC gives an accurate estimate of the true distribution of values

Learning Bayesian Learning Bayesian NetworksNetworks

Learning Bayesian networks

InducerInducerInducerInducerData + Prior information

E

R

B

A

C .9 .1

e

b

e

.7 .3

.99 .01

.8 .2

be

b

b

e

BE P(A | E,B)

Known Structure -- Complete Data

E, B, A<Y,N,N><Y,Y,Y><N,N,Y><N,Y,Y> . .<N,Y,Y>

InducerInducerInducerInducer

E B

A.9 .1

e

b

e

.7 .3

.99 .01

.8 .2

be

b

b

e

BE P(A | E,B)

? ?

e

b

e

? ?

? ?

? ?

be

b

b

e

BE P(A | E,B) E B

A

• Network structure is specified– Inducer needs to estimate parameters

• Data does not contain missing values

Unknown Structure -- Complete Data

E, B, A<Y,N,N><Y,Y,Y><N,N,Y><N,Y,Y> . .<N,Y,Y>


E B

A.9 .1

e

b

e

.7 .3

.99 .01

.8 .2

be

b

b

e

BE P(A | E,B)

? ?

e

b

e

? ?

? ?

? ?

be

b

b

e

BE P(A | E,B) E B

A

• Network structure is not specified– Inducer needs to select arcs & estimate parameters

• Data does not contain missing values

Known Structure -- Incomplete Data


E B

A.9 .1

e

b

e

.7 .3

.99 .01

.8 .2

be

b

b

e

BE P(A | E,B)

? ?

e

b

e

? ?

? ?

? ?

be

b

b

e

BE P(A | E,B) E B

A

• Network structure is specified• Data contains missing values

– We consider assignments to missing values

E, B, A<Y,N,N><Y,?,Y><N,N,Y><N,Y,?> . .<?,Y,Y>

Known Structure / Complete Data

• Given a network structure G– And choice of parametric family for P(Xi|Pai)

• Learn parameters for network

Goal• Construct a network that is “closest” to

probability that generated the data

Learning Parameters for a Bayesian Network

E B

A

C

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅⋅⋅⋅

=

][][][][

]1[]1[]1[]1[

MCMAMBME

CABE

D

• Training data has the form:


E B

A

C

• Since we assume i.i.d. samples,likelihood function is

∏ Θ=Θm

mCmAmBmEPDL ):][],[],[],[():(


E B

A

C

• By definition of network, we get

∏

∏

Θ

Θ

Θ

Θ

=

Θ=Θ

m

m

mAmCP

mEmBmAP

mBP

mEP

mCmAmBmEPDL

):][|][(

):][],[|][(

):][(

):][(

):][],[],[],[():(

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅⋅⋅⋅

][][][][

]1[]1[]1[]1[

MCMAMBME

CABE


E B

A

C

• Rewriting terms, we get

∏

∏

∏

∏

∏

Θ

Θ

Θ

Θ=

Θ=Θ

m

m

m

m

m

mAmCP

mEmBmAP

mBP

mEP

mCmAmBmEPDL

):][|][(

):][],[|][(

):][(

):][(

):][],[],[],[():(

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅⋅⋅⋅

][][][][

]1[]1[]1[]1[

MCMAMBME

CABE

General Bayesian Networks

Generalizing for any Bayesian network:

• The likelihood decomposes according to the structure of the network.

∏

∏∏

∏∏

∏

Θ=

Θ=

Θ=

Θ=Θ

iii

i miii

m iiii

mn

DL

mPamxP

mPamxP

mxmxPDL

):(

):][|][(

):][|][(

):][,],[():( 1 Ki.i.d. samples

Network factorization

General Bayesian Networks (Cont.)

Decomposition Independent Estimation Problems

If the parameters for each family are not related, then they can be estimated independently of

each other.

From Binomial to Multinomial

• For example, suppose X can have the values 1,2,…,K

• We want to learn the parameters 1, 2. …, K

Sufficient statistics:

• N1, N2, …, NK - the number of times each outcome is observed

Likelihood function:

MLE:

∏=

=K

k

Nk

kDL1

):(

∑=

llN

Nkk

ˆ

Likelihood for Multinomial Networks

• When we assume that P(Xi | Pai ) is multinomial, we get further decomposition:

∏∏

∏∏

∏ ∏

∏

=

Θ=

Θ=

Θ=Θ

=

i i

ii

ii

i i

ii

i ii

pa x

paxNpax

pa x

paxNiii

pa pamPamiii

miiiii

paxP

pamxP

mPamxPDL

),(|

),(

][,

):|(

):|][(

):][|][():(

θ

∏ Θ=Θm

iiiii mPamxPDL ):][|][():(

∏∏

∏ ∏

∏

Θ=

Θ=

Θ=Θ

=

i i

ii

i ii

pa x

paxNiii

pa pamPamiii

miiiii

paxP

pamxP

mPamxPDL

),(

][,

):|(

):|][(

):][|][():(

∏ ∏

∏

=

Θ=

Θ=Θ

i iipa pamPamiii

miiiii

pamxP

mPamxPDL

][,

):|][(

):][|][():(

Likelihood for Multinomial Networks

• When we assume that P(Xi | Pai ) is multinomial, we get further decomposition:

• For each value pai of the parents of Xi we get an independent multinomial problem

• The MLE is

∏∏=Θi i

ii

iipa x

paxNpaxii DL ),(

|):( θ

)(),(ˆ

|i

iipax paN

paxNii=

Maximum Likelihood Estimation

Consistency • Estimate converges to best possible value as the

number of examples grow

• To make this formal, we need to introduce some definitions

KL-Divergence

• Let P and Q be two distributions over X• A measure of distance between P and Q is the

Kullback-Leibler Divergence

• KL(P||Q) = 1 (when logs are in base 2) =– The probability P assigns to an instance is, on

average, half the probability Q assigns to it

• KL(P||Q) 0 • KL(P||Q) = 0 iff are P and Q equal

∑=x )x(Q

)x(Plog)x(P)Q||P(KL

Consistency

• Let P(X| ) be a parametric family– We need to make various regularity condition we won’t go

into now

• Let P*(X) be the distribution that generates the data

• Let be the MLE estimate given a dataset D

Thm• As N , where

with probability 1

D̂

*ˆ →D

):X(P||)X(P(KLminarg ** =

Consistency -- Geometric Interpretation

P*

P(X| * )

Space of probability distribution

Distributions that canrepresented by P(X| )

Bayesian Inference

Frequentist Approach:• Assumes there is an unknown but fixed parameter • Estimates with some confidence • Prediction by using the estimated parameter value

Bayesian Approach:• Represents uncertainty about the unknown

parameter• Uses probability to quantify this uncertainty:

– Unknown parameters as random variables

• Prediction follows from the rules of probability:– Expectation over the unknown parameters

Bayesian Inference (cont.)

• We can represent our uncertainty about the sampling process using a Bayesian network

• The values of X are independent given

• The conditional probabilities, P(x[m] | ), are the parameters in the model

• Prediction is now inference in this network

X[1] X[2] X[m] X[m+1]

Observed data Query

Bayesian Inference (cont.)

Prediction as inference in this network

where

+=

+=

+

∫∫

dMxxPMxP

dMxxPMxxMxP

MxxMxP

])[,],1[|()|]1[(

])[,],1[|(])[,],1[,|]1[(

])[,],1[|]1[(

K

KK

K

])[],1[()()|][],1[(

])[],1[|(MxxP

PMxxPMxxP

KK

K=

Posterior

Likelihood Prior

Probability of data

X[1] X[2] X[m] X[m+1]

Dirichlet Priors

• Recall that the likelihood function is

• A Dirichlet prior with hyperparameters 1,…,K is defined as

for legal 1,…, K

Then the posterior has the same form, with

hyperparameters 1+N 1,…,K +N K

∏=

=ΘK

1k

Nk

kDL θ):(

∏=

−∝ΘK

kk

kP1

1)(

∏∏∏=

−+

==

− =∝ΘΘ∝ΘK

k

Nk

K

k

Nk

K

kk

kkkkDPPDP1

1

11

1)|()()|( αα θθθ

Dirichlet Priors (cont.)

• We can compute the prediction on a new event in closed form:

• If P( ) is Dirichlet with hyperparameters 1,…,K then

• Since the posterior is also Dirichlet, we get

∑∫

=ΘΘ⋅==

ll

kk d)(P)k]1[X(P

∑∫ ++

=ΘΘ⋅==+

lll )N(

Nd)D|(P)D|k]1M[X(P kk

k

Prior Knowledge

• The hyperparameters 1,…,K can be thought of as “imaginary” counts from our prior experience

• Equivalent sample size = 1+…+K

• The larger the equivalent sample size the more confident we are in our prior

Bayesian Prediction(cont.)

• Given these observations, we can compute the posterior for each multinomial Xi | pai

independently– The posterior is Dirichlet with parameters

(Xi=1|pai)+N (Xi=1|pai),…, (Xi=k|pai)+N (Xi=k|pai)

• The predictive distribution is then represented by the parameters

)pa(N)pa()pa,x(N)pa,x(~

ii

iiiipa|x ii +

+=

Learning Parameters: Summary

• Estimation relies on sufficient statistics– For multinomial these are of the form N (xi,pai)

– Parameter estimation

• Bayesian methods also require choice of priors• Both MLE and Bayesian are asymptotically

equivalent and consistent• Both can be implemented in an on-line manner

by accumulating sufficient statistics

)()(),(),(~

|ii

iiiipax paNpa

paxNpaxii +

+=

)(),(ˆ

|i

iipax paN

paxNii=

MLE Bayesian (Dirichlet)

Learning Structure from Complete Data

Why Struggle for Accurate Structure?

• Increases the number of parameters to be fitted

• Wrong assumptions about causality and domain structure

• Cannot be compensated by accurate fitting of parameters

• Also misses causality and domain structure

Earthquake Alarm Set

Sound

BurglaryEarthquake Alarm Set

Sound

Burglary

Earthquake Alarm Set

Sound

Burglary

Adding an arc Missing an arc

Approaches to Learning Structure

• Constraint based– Perform tests of conditional independence– Search for a network that is consistent with the

observed dependencies and independencies

• Pros & Cons Intuitive, follows closely the construction of BNs Separates structure learning from the form of the

independence tests Sensitive to errors in individual tests

Approaches to Learning Structure

• Score based– Define a score that evaluates how well the (in)dependencies

in a structure match the observations– Search for a structure that maximizes the score

• Pros & Cons Statistically motivated Can make compromises Takes the structure of conditional probabilities into account Computationally hard

Likelihood Score for Structures

First cut approach: – Use likelihood function

• Recall, the likelihood score for a network structure and parameters is

• Since we know how to maximize parameters from now we assume

∏∏

∏Θ=

Θ=Θ

m ii,G

Gii

mGn1G

),G:]m[Pa|]m[x(P

),G:]m[x,],m[x(P)D:,G(L K

):,(max):( DGLDGL GGΘ= Θ

Avoiding Overfitting

“Classic” issue in learning. Approaches:• Restricting the hypotheses space

– Limits the overfitting capability of the learner– Example: restrict # of parents or # of parameters

• Minimum description length– Description length measures complexity– Prefer models that compactly describes the training data

• Bayesian methods– Average over all possible parameter values– Use prior knowledge

Bayesian Inference

• Bayesian Reasoning---compute expectation over unknown G

• Assumption: Gs are mutually exclusive and exhaustive

• We know how to compute P(x[M+1]|G,D)– Same as prediction with fixed structure

• How do we compute P(G|D)?

∑ +=+G

)D|G(P)G,D|]1M[x(P)D|]1M[x(P

Marginal likelihood

Prior over structures

)()()|(

)|(DP

GPGDPDGP =

Using Bayes rule:

P(D) is the same for all structures GCan be ignored when comparing structures

Probability of Data

Posterior Score

Marginal Likelihood

• By introduction of variables, we have that

• This integral measures sensitivity to choice of parameters

∫= dGPGDPGDP )|(),|()|(

LikelihoodPrior over parameters

Marginal Likelihood: Multinomials

The same argument generalizes to multinomials with Dirichlet prior

• P( ) is Dirichlet with hyperparameters 1,…,K

• D is a dataset with sufficient statistics N1,…,NK

Then

∏∑

∑Γ

+Γ

⎟⎠

⎞⎜⎝

⎛+Γ

⎟⎠

⎞⎜⎝

⎛Γ

=l l

ll

lll

ll

)()(

)()(

αα

α

αN

NDP

Marginal Likelihood for General Network

The marginal likelihood has the form:

where • N(..) are the counts from the data (..) are the hyperparameters for each family

given G

( )( )∏∏ ∏ Γ

+Γ

+Γ

Γ=

i pa xG

i

Gi

Gi

GG

G

Gi i i

ii

ii

i

pax

paxNpax

paNpa

paGDP

)),((

)),(),((

)()(

)()|(

α

α

α

α

Dirichlet Marginal LikelihoodFor the sequence of values of Xi when

Xi’s parents have a particular value

Priors

• We need: prior counts (..) for each network structure G

• This can be a formidable task– There are exponentially many structures…

BDe Score

Possible solution: The BDe prior

• Represent prior using two elements M0, B0

– M0 - equivalent sample size

– B0 - network representing the prior probability of events

BDe Score

Intuition: M0 prior examples distributed by B0

• Set (xi,paiG) = M0 P(xi,pai

G| B0) – Note that pai

G are not the same as the parents of Xi in B0.

– Compute P(xi,paiG| B0) using standard inference

procedures

• Such priors have desirable theoretical properties– Equivalent networks are assigned the same score

Bayesian Score: Asymptotic Behavior

Theorem: If the prior P(Θ |G) is “well-behaved”, then

)1()dim(2

log):()|(log OG

MDGlGDP +−=

Asymptotic Behavior: Consequences

• Bayesian score is consistent– As M the “true” structure G* maximizes the score

(almost surely)– For sufficiently large M, the maximal scoring

structures are equivalent to G*

• Observed data eventually overrides prior information– Assuming that the prior assigns positive probability to

all cases

)1()dim(2

log):()|(log OG

MDGlGDP +−=

Asymptotic Behavior

• This score can also be justified by the Minimal Description Length (MDL) principle

• This equation explicitly shows the tradeoff between– Fitness to data --- likelihood term– Penalty for complexity --- regularization term

)dim(2

log):():Score( G

MDGlDG −=

Scores -- Summary

• Likelihood, MDL, (log) BDe have the form

• BDe requires assessing prior network.It can naturally incorporate prior knowledge and previous experience

• BDe is consistent and asymptotically equivalent (up to a constant) to MDL

• All are score-equivalent– G equivalent to G’ Score(G) = Score(G’)

))(:|():( iii

Gi PaXNPaXScoreDGScore

i∑=

Optimization Problem

Input:– Training data– Scoring function (including priors, if needed)– Set of possible structures

• Including prior knowledge about structure

Output:– A network (or networks) that maximize the score

Key Property:– Decomposability: the score of a network is a sum of

terms.

Difficulty

Theorem: Finding maximal scoring network structure with at most k parents for each variables is NP-hard for k > 1

Heuristic Search

We address the problem by using heuristic search

• Define a search space:– nodes are possible structures– edges denote adjacency of structures

• Traverse this space looking for high-scoring structures

Search techniques:– Greedy hill-climbing– Best first search– Simulated Annealing– ...

Heuristic Search (cont.)

• Typical operations:

S C

E

D

S C

E

D

Reverse C EDelete C

E

Add C

D

S C

E

D

S C

E

D

Exploiting Decomposability in Local Search

• Caching: To update the score of after a local change, we only need to re-score the families that were changed in the last move

S C

E

D

S C

E

D

S C

E

D

S C

E

D

Greedy Hill-Climbing

• Simplest heuristic local search– Start with a given network

• empty network• best tree • a random network

– At each iteration• Evaluate all possible changes• Apply change that leads to best improvement in score• Reiterate

– Stop when no modification improves score

• Each step requires evaluating approximately n new changes

Greedy Hill-Climbing: Possible Pitfalls

• Greedy Hill-Climbing can get struck in:– Local Maxima:

• All one-edge changes reduce the score

– Plateaus:• Some one-edge changes leave the score unchanged• Happens because equivalent networks received the

same score and are neighbors in the search space

• Both occur during structure search• Standard heuristics can escape both

– Random restarts– TABU search

Model Selection

• So far, we focused on single model– Find best scoring model– Use it to predict next example

• Implicit assumption: – Best scoring model dominates the weighted

sum

• Pros:– We get a single structure– Allows for efficient use in our tasks

• Cons:– We are committing to the independencies of a

particular structure– Other structures might be as probable given the data

Model Averaging

• Recall, Bayesian analysis started with

– This requires us to average over all possible models

∑ +=+G

DGPGDMxPDMxP )|(),|]1[()|]1[(

Model Averaging (cont.)

• Full Averaging– Sum over all structures– Usually intractable---there are exponentially many structures

• Approximate Averaging– Find K largest scoring structures– Approximate the sum by averaging over their prediction– Weight of each structure determined by the Bayes Factor

)()(

)'|()'()|()(

)|'()|(

DPDP

GDPGPGDPGP

DGPDGP

⋅=

The actual score we compute

Search: Summary

• Discrete optimization problem• In general, NP-Hard

– Need to resort to heuristic search– In practice, search is relatively fast (~100 vars in ~10 min):

• Decomposability• Sufficient statistics

• In some cases, we can reduce the search problem to an easy optimization problem– Example: learning trees

describing data

Documents

values of y

values x

y z example

variables x1

independent ifpx

joint distribution of

x5 independent

independent given z